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Theorem lmhmplusg 16120
Description: The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypothesis
Ref Expression
lmhmplusg.p  |-  .+  =  ( +g  `  N )
Assertion
Ref Expression
lmhmplusg  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M LMHom  N
) )

Proof of Theorem lmhmplusg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . 2  |-  ( Base `  M )  =  (
Base `  M )
2 eqid 2436 . 2  |-  ( .s
`  M )  =  ( .s `  M
)
3 eqid 2436 . 2  |-  ( .s
`  N )  =  ( .s `  N
)
4 eqid 2436 . 2  |-  (Scalar `  M )  =  (Scalar `  M )
5 eqid 2436 . 2  |-  (Scalar `  N )  =  (Scalar `  N )
6 eqid 2436 . 2  |-  ( Base `  (Scalar `  M )
)  =  ( Base `  (Scalar `  M )
)
7 lmhmlmod1 16109 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  M  e.  LMod )
87adantr 452 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  M  e.  LMod )
9 lmhmlmod2 16108 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  N  e.  LMod )
109adantr 452 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  N  e.  LMod )
114, 5lmhmsca 16106 . . 3  |-  ( F  e.  ( M LMHom  N
)  ->  (Scalar `  N
)  =  (Scalar `  M ) )
1211adantr 452 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  (Scalar `  N
)  =  (Scalar `  M ) )
13 lmodabl 15991 . . . 4  |-  ( N  e.  LMod  ->  N  e. 
Abel )
1410, 13syl 16 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  N  e.  Abel )
15 lmghm 16107 . . . 4  |-  ( F  e.  ( M LMHom  N
)  ->  F  e.  ( M  GrpHom  N ) )
1615adantr 452 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  F  e.  ( M  GrpHom  N ) )
17 lmghm 16107 . . . 4  |-  ( G  e.  ( M LMHom  N
)  ->  G  e.  ( M  GrpHom  N ) )
1817adantl 453 . . 3  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  G  e.  ( M  GrpHom  N ) )
19 lmhmplusg.p . . . 4  |-  .+  =  ( +g  `  N )
2019ghmplusg 15461 . . 3  |-  ( ( N  e.  Abel  /\  F  e.  ( M  GrpHom  N )  /\  G  e.  ( M  GrpHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M  GrpHom  N ) )
2114, 16, 18, 20syl3anc 1184 . 2  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M  GrpHom  N ) )
22 simpll 731 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F  e.  ( M LMHom  N ) )
23 simprl 733 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  x  e.  ( Base `  (Scalar `  M ) ) )
24 simprr 734 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  y  e.  ( Base `  M
) )
254, 6, 1, 2, 3lmhmlin 16111 . . . . . 6  |-  ( ( F  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( F `  (
x ( .s `  M ) y ) )  =  ( x ( .s `  N
) ( F `  y ) ) )
2622, 23, 24, 25syl3anc 1184 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( F `  ( x
( .s `  M
) y ) )  =  ( x ( .s `  N ) ( F `  y
) ) )
27 simplr 732 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G  e.  ( M LMHom  N ) )
284, 6, 1, 2, 3lmhmlin 16111 . . . . . 6  |-  ( ( G  e.  ( M LMHom 
N )  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( G `  (
x ( .s `  M ) y ) )  =  ( x ( .s `  N
) ( G `  y ) ) )
2927, 23, 24, 28syl3anc 1184 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( G `  ( x
( .s `  M
) y ) )  =  ( x ( .s `  N ) ( G `  y
) ) )
3026, 29oveq12d 6099 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F `  (
x ( .s `  M ) y ) )  .+  ( G `
 ( x ( .s `  M ) y ) ) )  =  ( ( x ( .s `  N
) ( F `  y ) )  .+  ( x ( .s
`  N ) ( G `  y ) ) ) )
319ad2antrr 707 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  N  e.  LMod )
3211fveq2d 5732 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  M )
) )
3332ad2antrr 707 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( Base `  (Scalar `  N
) )  =  (
Base `  (Scalar `  M
) ) )
3423, 33eleqtrrd 2513 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  x  e.  ( Base `  (Scalar `  N ) ) )
35 eqid 2436 . . . . . . . 8  |-  ( Base `  N )  =  (
Base `  N )
361, 35lmhmf 16110 . . . . . . 7  |-  ( F  e.  ( M LMHom  N
)  ->  F :
( Base `  M ) --> ( Base `  N )
)
3736ad2antrr 707 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F : ( Base `  M
) --> ( Base `  N
) )
3837, 24ffvelrnd 5871 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( F `  y )  e.  ( Base `  N
) )
391, 35lmhmf 16110 . . . . . . 7  |-  ( G  e.  ( M LMHom  N
)  ->  G :
( Base `  M ) --> ( Base `  N )
)
4039ad2antlr 708 . . . . . 6  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G : ( Base `  M
) --> ( Base `  N
) )
4140, 24ffvelrnd 5871 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( G `  y )  e.  ( Base `  N
) )
42 eqid 2436 . . . . . 6  |-  ( Base `  (Scalar `  N )
)  =  ( Base `  (Scalar `  N )
)
4335, 19, 5, 3, 42lmodvsdi 15973 . . . . 5  |-  ( ( N  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  N )
)  /\  ( F `  y )  e.  (
Base `  N )  /\  ( G `  y
)  e.  ( Base `  N ) ) )  ->  ( x ( .s `  N ) ( ( F `  y )  .+  ( G `  y )
) )  =  ( ( x ( .s
`  N ) ( F `  y ) )  .+  ( x ( .s `  N
) ( G `  y ) ) ) )
4431, 34, 38, 41, 43syl13anc 1186 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  N ) ( ( F `  y ) 
.+  ( G `  y ) ) )  =  ( ( x ( .s `  N
) ( F `  y ) )  .+  ( x ( .s
`  N ) ( G `  y ) ) ) )
4530, 44eqtr4d 2471 . . 3  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F `  (
x ( .s `  M ) y ) )  .+  ( G `
 ( x ( .s `  M ) y ) ) )  =  ( x ( .s `  N ) ( ( F `  y )  .+  ( G `  y )
) ) )
46 ffn 5591 . . . . 5  |-  ( F : ( Base `  M
) --> ( Base `  N
)  ->  F  Fn  ( Base `  M )
)
4737, 46syl 16 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  F  Fn  ( Base `  M
) )
48 ffn 5591 . . . . 5  |-  ( G : ( Base `  M
) --> ( Base `  N
)  ->  G  Fn  ( Base `  M )
)
4940, 48syl 16 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  G  Fn  ( Base `  M
) )
50 fvex 5742 . . . . 5  |-  ( Base `  M )  e.  _V
5150a1i 11 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  ( Base `  M )  e. 
_V )
527ad2antrr 707 . . . . 5  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  M  e.  LMod )
531, 4, 2, 6lmodvscl 15967 . . . . 5  |-  ( ( M  e.  LMod  /\  x  e.  ( Base `  (Scalar `  M ) )  /\  y  e.  ( Base `  M ) )  -> 
( x ( .s
`  M ) y )  e.  ( Base `  M ) )
5452, 23, 24, 53syl3anc 1184 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  M ) y )  e.  ( Base `  M
) )
55 fnfvof 6317 . . . 4  |-  ( ( ( F  Fn  ( Base `  M )  /\  G  Fn  ( Base `  M ) )  /\  ( ( Base `  M
)  e.  _V  /\  ( x ( .s
`  M ) y )  e.  ( Base `  M ) ) )  ->  ( ( F  o F  .+  G
) `  ( x
( .s `  M
) y ) )  =  ( ( F `
 ( x ( .s `  M ) y ) )  .+  ( G `  ( x ( .s `  M
) y ) ) ) )
5647, 49, 51, 54, 55syl22anc 1185 . . 3  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  o F 
.+  G ) `  ( x ( .s
`  M ) y ) )  =  ( ( F `  (
x ( .s `  M ) y ) )  .+  ( G `
 ( x ( .s `  M ) y ) ) ) )
57 fnfvof 6317 . . . . 5  |-  ( ( ( F  Fn  ( Base `  M )  /\  G  Fn  ( Base `  M ) )  /\  ( ( Base `  M
)  e.  _V  /\  y  e.  ( Base `  M ) ) )  ->  ( ( F  o F  .+  G
) `  y )  =  ( ( F `
 y )  .+  ( G `  y ) ) )
5847, 49, 51, 24, 57syl22anc 1185 . . . 4  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  o F 
.+  G ) `  y )  =  ( ( F `  y
)  .+  ( G `  y ) ) )
5958oveq2d 6097 . . 3  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
x ( .s `  N ) ( ( F  o F  .+  G ) `  y
) )  =  ( x ( .s `  N ) ( ( F `  y ) 
.+  ( G `  y ) ) ) )
6045, 56, 593eqtr4d 2478 . 2  |-  ( ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  /\  (
x  e.  ( Base `  (Scalar `  M )
)  /\  y  e.  ( Base `  M )
) )  ->  (
( F  o F 
.+  G ) `  ( x ( .s
`  M ) y ) )  =  ( x ( .s `  N ) ( ( F  o F  .+  G ) `  y
) ) )
611, 2, 3, 4, 5, 6, 8, 10, 12, 21, 60islmhmd 16115 1  |-  ( ( F  e.  ( M LMHom 
N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M LMHom  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   .scvsca 13533    GrpHom cghm 15003   Abelcabel 15413   LModclmod 15950   LMHom clmhm 16095
This theorem is referenced by:  nmhmplusg  18791  mendrng  27477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-ghm 15004  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lmhm 16098
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